Problem 68 - Intermediate Algebra

Question

Simplify by combining like terms. See Example 5 . $$3.7 x^{2}+3.3 x^{2}-1.1 x^{2}$$

Step-by-Step Solution

Answer

The simplified expression is \(5.9x^2\).

Step 1: Identify Like Terms

Examine the expression \(3.7x^2 + 3.3x^2 - 1.1x^2\). All parts of this expression have the term \(x^2\) with different coefficients. This means they are like terms since they all share the same variable form \(x^2\).

Step 2: Combine the Coefficients

Add and subtract the coefficients of the like terms \(3.7\), \(3.3\), and \(-1.1\). Calculate: \(3.7 + 3.3 - 1.1 = 5.9\).

Step 3: Write the Simplified Expression

Combine the results from the previous step with the common term \(x^2\). The simplified expression is \(5.9x^2\).

Key Concepts

Simplifying Expressions

Simplifying expressions is a fundamental skill in algebra that involves making expressions easier to work with by reducing them to their most compact form. This often involves combining like terms, which are terms that have the same variable raised to the same power.
For example, in the expression given: \(3.7x^2 + 3.3x^2 - 1.1x^2\), each term includes the variable \(x^2\). Here, simplifying means to group and add or subtract these like terms together.
When you simplify an expression, you reduce the total number of terms without changing the value of the expression. This not only makes the expression simpler, but also easier to understand and solve, especially in larger equations.
This concept is particularly useful when dealing with equations or expressions that involve multiple variables or long polynomials, allowing you to streamline calculations and find solutions more efficiently.

Coefficients

In algebra, a coefficient is the numerical part of a term that contains a variable. It is the number that is multiplied by the variable in any given term.
In our expression \(3.7x^2 + 3.3x^2 - 1.1x^2\), the coefficients are 3.7, 3.3, and -1.1 respectively. Each of these numbers is connected to the \(x^2\) term.
Understanding coefficients is crucial when simplifying expressions because they tell you how much of one variable you have. When combining like terms, you are essentially adding or subtracting these coefficients together.

• Positive coefficients increase the value of the expression as they add more of a particular term.
• Negative coefficients reduce the value or take away a portion of a term when simplified.

By focusing on the coefficients, you can quickly manage and simplify complex expressions by altering only the numerical values while keeping the variables intact.

Polynomial Expressions

Polynomial expressions are mathematical expressions involving sums and differences of terms with variables raised to whole-number exponents. They provide a broad framework for handling algebraic calculations and are classified based on their degree—defined by the highest power of the variable present.
In the exercise \(3.7x^2 + 3.3x^2 - 1.1x^2\), we have a polynomial expression, notably a quadratic polynomial since the highest exponent is 2. This expression is composed of several terms, which here are like terms, making it straightforward to simplify.
Working with polynomial expressions often involves identifying similar terms, understanding the role of coefficients, and using operations like addition and subtraction to simplify them. This is vital in solving equations, factoring expressions, and understanding the relationships and patterns within algebraic problems.

• Lower-degree polynomials (linear or quadratic) often provide a stepping stone to more complex algebraic concepts.
• Combining like terms and understanding the structure of polynomials can also aid in graphing them and in solving more advanced mathematical problems.

Overall, managing polynomial expressions effectively is key to advancing in algebra and tackling more intricate mathematical challenges.